Volume 28, Issue 5
Numerical Simulations for Full History Recursive Multilevel Picard Approximations for Systems of High-Dimensional Partial Differential Equations

Commun. Comput. Phys., 28 (2020), pp. 2109-2138.

Published online: 2020-11

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• Abstract

One of the most challenging issues in applied mathematics is to develop and analyze algorithms which are able to approximately compute solutions of high-dimensional nonlinear partial differential equations (PDEs). In particular, it is very hard to develop approximation algorithms which do not suffer under the curse of dimensionality in the sense that the number of computational operations needed by the algorithm to compute an approximation of accuracy $ε$>0 grows at most polynomially in both the reciprocal 1/$ε$ of the required accuracy and the dimension $d∈\mathbb{N}$ of the PDE. Recently, a new approximation method, the so-called full history recursive multilevel Picard (MLP) approximation method, has been introduced and, until today, this approximation scheme is the only approximation method in the scientific literature which has been proven to overcome the curse of dimensionality in the numerical approximation of semilinear PDEs with general time horizons. It is a key contribution of this article to extend the MLP approximation method to systems of semilinear PDEs and to numerically test it on several example PDEs. More specifically, we apply the proposed MLP approximation method in the case of Allen-Cahn PDEs, Sine-Gordon-type PDEs, systems of coupled semilinear heat PDEs, and semilinear Black-Scholes PDEs in up to 1000 dimensions. We also compare the performance of the proposed MLP approximation algorithm with a deep learning based approximation method from the scientific literature.

• Keywords

Curse of dimensionality, partial differential equation, PDE, high-dimensional, semilinear, multilevel Picard.

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@Article{CiCP-28-2109, author = {Becker , Sebastian and Braunwarth , Ramon and Hutzenthaler , Martin and Jentzen , Arnulf and von Wurstemberger , Philippe}, title = {Numerical Simulations for Full History Recursive Multilevel Picard Approximations for Systems of High-Dimensional Partial Differential Equations}, journal = {Communications in Computational Physics}, year = {2020}, volume = {28}, number = {5}, pages = {2109--2138}, abstract = {

One of the most challenging issues in applied mathematics is to develop and analyze algorithms which are able to approximately compute solutions of high-dimensional nonlinear partial differential equations (PDEs). In particular, it is very hard to develop approximation algorithms which do not suffer under the curse of dimensionality in the sense that the number of computational operations needed by the algorithm to compute an approximation of accuracy $ε$>0 grows at most polynomially in both the reciprocal 1/$ε$ of the required accuracy and the dimension $d∈\mathbb{N}$ of the PDE. Recently, a new approximation method, the so-called full history recursive multilevel Picard (MLP) approximation method, has been introduced and, until today, this approximation scheme is the only approximation method in the scientific literature which has been proven to overcome the curse of dimensionality in the numerical approximation of semilinear PDEs with general time horizons. It is a key contribution of this article to extend the MLP approximation method to systems of semilinear PDEs and to numerically test it on several example PDEs. More specifically, we apply the proposed MLP approximation method in the case of Allen-Cahn PDEs, Sine-Gordon-type PDEs, systems of coupled semilinear heat PDEs, and semilinear Black-Scholes PDEs in up to 1000 dimensions. We also compare the performance of the proposed MLP approximation algorithm with a deep learning based approximation method from the scientific literature.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0130}, url = {http://global-sci.org/intro/article_detail/cicp/18406.html} }
TY - JOUR T1 - Numerical Simulations for Full History Recursive Multilevel Picard Approximations for Systems of High-Dimensional Partial Differential Equations AU - Becker , Sebastian AU - Braunwarth , Ramon AU - Hutzenthaler , Martin AU - Jentzen , Arnulf AU - von Wurstemberger , Philippe JO - Communications in Computational Physics VL - 5 SP - 2109 EP - 2138 PY - 2020 DA - 2020/11 SN - 28 DO - http://doi.org/10.4208/cicp.OA-2020-0130 UR - https://global-sci.org/intro/article_detail/cicp/18406.html KW - Curse of dimensionality, partial differential equation, PDE, high-dimensional, semilinear, multilevel Picard. AB -

One of the most challenging issues in applied mathematics is to develop and analyze algorithms which are able to approximately compute solutions of high-dimensional nonlinear partial differential equations (PDEs). In particular, it is very hard to develop approximation algorithms which do not suffer under the curse of dimensionality in the sense that the number of computational operations needed by the algorithm to compute an approximation of accuracy $ε$>0 grows at most polynomially in both the reciprocal 1/$ε$ of the required accuracy and the dimension $d∈\mathbb{N}$ of the PDE. Recently, a new approximation method, the so-called full history recursive multilevel Picard (MLP) approximation method, has been introduced and, until today, this approximation scheme is the only approximation method in the scientific literature which has been proven to overcome the curse of dimensionality in the numerical approximation of semilinear PDEs with general time horizons. It is a key contribution of this article to extend the MLP approximation method to systems of semilinear PDEs and to numerically test it on several example PDEs. More specifically, we apply the proposed MLP approximation method in the case of Allen-Cahn PDEs, Sine-Gordon-type PDEs, systems of coupled semilinear heat PDEs, and semilinear Black-Scholes PDEs in up to 1000 dimensions. We also compare the performance of the proposed MLP approximation algorithm with a deep learning based approximation method from the scientific literature.

Sebastian Becker, Ramon Braunwarth, Martin Hutzenthaler, Arnulf Jentzen & Philippe von Wurstemberger. (2020). Numerical Simulations for Full History Recursive Multilevel Picard Approximations for Systems of High-Dimensional Partial Differential Equations. Communications in Computational Physics. 28 (5). 2109-2138. doi:10.4208/cicp.OA-2020-0130
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